Question

Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A coin with (random) bias P (i.e., Pr[H] = P) is flipped three times. Assume that the value of P does not change during the sequence of tosses. a. What is the probability that all three flips are heads? b. Find the probability that the second flip is heads given that the first flip is heads. c. Is the second flip independent of the first flip?

Answer 1

Answer:-

Given that:-

Given that P be a random variable distributed uniformly over[0,1] . A coin with bias P is flipped three times.

a.Let $X_{i}$ be outcome of the flip a ith coin where i=1,2,3

The probability that all three flips are heads is

$P(X_{1}=H,X_{2}=H,X_{3}=H)=\int_{0}^{1}P(X_{1}=H,X_{2}=H,X_{3}=H\setminus P=p)fp(p)dp$

IN the question , the value of P does not change during the sequence of tosses, So

P(Xi = 1, X3 = 41, X3 = H) = [P(X; H), P(X2 = H), P(X3 H) fp(p)dp
$P(X_{1}=H,X_{2}=H,X_{3}=H)=\int_{0}^{1}p^{3}dp$

P(X1 = H, X2 = H, X3 = H) [1]

$P(X_{1}=H,X_{2}=H,X_{3}=H)=\frac{1}{4}$

b. The probability that the second flip is heads given that the first flip is heads

$P(X_{2}=H\setminus X_{1}=H)=\frac{P(X_{2}=H,X_{1}=H)}{P(X_{1}=H)}$

$P( X_{1}=H)=\int_{0}^{1}P(X_{1}=H\setminus P=p)fp(p)dp$

$P( X_{1}=H)=\int_{0}^{1}pdp=\frac{1}{2}$

and

$P(X_{1}=H,X_{2}=H)=\int_{0}^{1}P(X_{1}=H,X_{2}=H\setminus P=p)fp(p)dp$$P(X_{1}=H,X_{2}=H)=\int_{0}^{1}p^{2}fp(p)dp$

$P(X_{1}=H,X_{2}=H)=\int_{0}^{1}p^{2}dp=\frac{1}{3}$

$P(X_{2}=H\setminus X_{1}=H)=\frac{2}{3}$

c.For independent,

$P(X_{2}=H, X_{1}=H)=\frac{1}{3}$

$P(X_{2}=H)=P(X_{1}=H)=\frac{1}{2}$

This show that

$P(X_{2}=H,X_{1}=H)\neq P(X_{2}=H)P(X_{1}=H)$

The second flip is not independent of the first flip.